Safety-Critical Control under Uncertainty via Adaptive Conformal Quantile Prediction Intervals

Hao Zhou; Yanze Zhang; Yiwei Lyu; Wenhao Luo

cs.ROSep 20, 2026

Safety-Critical Control under Uncertainty via Adaptive Conformal Quantile Prediction Intervals

Authors: Hao Zhou, Yanze Zhang, Yiwei Lyu, Wenhao Luo

Organizations: Department of Computer Science, University of Illinois Chicago, Chicago, IL, 60607, USA · Department of Computer Science and Engineering, Texas A&M University, College Station, TX, 77843, USA

Abstract

Safety-critical control under uncertainty requires uncertainty representations that are both statistically valid (for certifiable performance) and compatible with enforceable safety constraints. However, existing methods often assume particular distributions of uncertainty for provable safety guarantees or establish symmetric and input-agnostic prediction intervals for robust safety, which can lead to misaligned or overly conservative safety constraints in control synthesis. In this paper, we introduce a novel safe control framework with adaptive uncertainty quantification that constructs calibrated and state-dependent prediction intervals to enable high-probability safety guarantees, while improving constrained control performance. The framework leverages adaptive conformal prediction (ACP) and extends it with conformal quantile regression (CQR) to capture distribution-free, asymmetric uncertainty intervals with certifiable probabilistic coverage, and integrates the resulting uncertainty sets into a probabilistic control barrier function formulation to enforce robust safety with reduced conservativeness. This yields uncertainty-aware safe control constraints that can be incorporated within a model predictive control(MPC) framework to provide provably safe behaviors with high probability. Simulation and theoretical results are provided to demonstrate the effectiveness of our approach.

Figures & tables

Fig. 1: CQR score computation under MPC. Considering the prediction horizon $H=3$ at time step $t_{k-1}$ . Let $F_{t_{k-1-\tau}}=F(\bar{x}_{k-1|k-1-\tau},u_{k-1|k-1-\tau})$ , $\hat{F}^{l}_{t_{k-1}}=\hat{F}_{\alpha_{l}}(\hat{x}_{k-1|k-1},u_{k-1|k-1};d_{\alpha_{l}})$ , and $\hat{F}^{u}_{t_{k-1}}=\hat{F}_{\alpha_{u}}(\hat{x}_{k-1|k-1},u_{k-1|k-1};d_{\alpha_{u}})$ . The time-lagged CQR score is defined as $S_{k-1+\tau|k-1}=\mathrm{max}(\hat{F}^{l}_{t_{k-1}}-F_{t_{k-1-\tau}},F_{t_{k-1-\tau}}+\hat{F}^{u}_{t_{k-1}})$ for $\tau=0,1,\dots,H-1$ . The resulting error terms $S_{k-1+\tau|k-1}$ are then added into the conformal score set $\mathcal{S}$ .

Fig. 2: Residual score computation with DT-CBF under MPC. For the current prediction horizon ( $H=3$ ) starting at time step $t_{k-1}$ , the time-lagged CBF residual is defined as $S^{\mathrm{rs}}_{k-1+\tau|k-1}=|F(\hat{x}_{k-1|k-1},u_{k-1|k-1})-F(\bar{x}_{k-1|k-1-\tau},u_{k-1|k-1-\tau})|$ for $\tau=1,\dots,H-1$ . For $\tau=0$ , the residual reduces to $S^{rs}_{k-1|k-1}=|F_{t_{k-1}}-F_{t_{k}}|$ . The resulting error terms ${S^{rs}_{k+\tau|k}}$ are then aggregated into the conformal score set $\mathcal{S}$ .

		Mobile Robot			Quadrotor
	Distribution	MC	MCA	MCA-CQR(ours)	MC	MCA	MCA-CQR(ours)
	Uniform	0	100	100	45	100	100
%Succ	Gaussian	0	100	100	41	100	100
	Time Varying	0	100	100	40	100	100
	Uniform	-0.026	0.027	0.008	-0.056	0.102	0.054
$\min h$	Gaussian	-0.128	0.032	0.006	-0.054	0.086	0.057

TABLE I: Quantitative experiment results under different distributions and different robot platforms

Panel of Figure 4 — Fig. 3: Simulations for a mobile robot navigating in a single obstacle scenario under different noise distributions with different methods. $\gamma=0.9$ and $\alpha=0.05$ are adopted. Our proposed framework (MCA-CQR) not only achieves safe navigation across different motion noise distributions but also provides a smaller safety margin than the MCA.

Panel of Figure 5 — Fig. 4: Simulations for a planar quadrotor navigating in a single obstacle scenario under variant noise distributions with different methods. $\gamma=0.9$ and $\alpha=0.05$ are adopted. Our proposed framework (MCA-CQR) not only achieves safe navigation across different motion noise distributions but also provides a smaller safety margin than the MCA.

Fig. 5: Parameter analysis with MC [ 22 ] and MCA-CQR(ours) under different MPC horizon

H

and the

\gamma

in CBF. This is conducted under the mobile robot platform. MC collides with the obstacle across different horizons, even with a small

\gamma

, whereas our method can achieve a collision-free path. In this simulation, we apply the Gaussian distribution or the uniform distribution randomly to

\epsilon_{k}

at every step

k

Fig. 6: Simulation for a planar quadrotor to track a circle while avoiding the obstacle. MC, MCA, and MCA-CQR are all applied. Results show that MC collides with the obstacle, whereas the uncertainty-aware method proposed in this paper (MCA-CQR) not only guarantees safety but also achieves a smaller safety margin than MCA under the residual score function.

Panel of Figure 8 — Fig. 7: Simulations for a quadrotor to track a figure eight while avoiding the obstacle under the time-varying motion noise. $\gamma=0.9$ and $\alpha=0.05$ are adopted. Our proposed framework (MCA-CQR) not only guarantees safety but also provides a small safety margin with respect to the MCA under the residual score function.

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Conformal Prediction Adaptive Conformal Inference Safety-Critical Control

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Safety-critical control of uncertain, adaptive systems often relies on conservative, worst-case uncertainty bounds that limit closed-loop performance. Online conformal prediction is a powerful data-driven method for quantifying uncertainty when truth values of predicted outputs are revealed online; however, for systems that adapt the dynamics without measurements of the state derivatives, standard online conformal prediction is insufficient to quantify the model uncertainty. We propose Staggered Integral Online Conformal Prediction (SI-OCP), an algorithm utilizing an integral score function to quantify the lumped effect of disturbance and learning error. This approach provides long-run coverage guarantees, resulting in long-run safety when synthesized with safety-critical controllers, including robust tube model predictive control. Finally, we validate the proposed approach through a numerical simulation of an all-layer deep neural network (DNN) adaptive quadcopter using robust tube (MPC), highlighting the applicability of our method to complex learning parameterizations and control strategies.

Abstract

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